We investigate how the stellar and gas-phase He abundances evolve as functions of time within simulated star-forming disc galaxies with different star formation histories. We make use of a cosmological chemodynamical simulation for galaxy formation and evolution, which includes star formation, as well as energy and chemical enrichment feedback from asymptotic giant branch stars, core-collapse supernovae, and Type Ia supernovae. The predicted relations between the He mass fraction, $Y$, and the metallicity, $Z$, in the interstellar medium of our simulated disc galaxies depend on the past galaxy star formation history. In particular, $dY/dZ$ is not constant and evolves as a function of time, depending on the specific chemical element that we choose to trace $Z$; in particular, $dY/dX_{text{O}}$ and $dY/dX_{text{C}}$ increase as functions of time, whereas $dY/dX_{text{N}}$ decreases. In the gas-phase, we find negative radial gradients of $Y$, due to the inside-out growth of our simulated galaxy discs as a function of time; this gives rise to longer chemical enrichment time scales in the outer galaxy regions, where we find lower average values for $Y$ and $Z$. Finally, by means of chemical evolution models, in the galactic bulge and inner disc, we predict steeper $Y$ versus age relations at high $Z$ than in the outer galaxy regions. We conclude that, for calibrating the assumed $Y$-$Z$ relation in stellar models, C, N, and C+N are better proxies for the metallicity than O, because they show steeper and less scattered relations.
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